Partial duality and closed 2-cell embeddings

Abstract

In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell.

New Dualities From Old: generating geometric, Petrie, and Wilson dualities and trialites of ribbon graphs

“…Operators on embedded graphs or subsets of their edges. [6,17,18]. However, the ribbon group action of [7,8] is limited in that self twuality properties under it are restricted to only the case of canonical self-twuality, in which the canonical identification of the edges of the graph and its twual yields an isomorphism.…”

“…In one direction there has been interest in the forms of partial duals of general graphs, particularly their genus(es) as well as when the partial duals are Eulerian or bipartite. For instance, Ellingham and Zha investigate when a partial dual has the property that every face is bounded by a cycle [6], Huggett and Moffat characterize when partial duals of plane graphs are bipartite [11], and Metsidik and Jin characterize when partial duals of plane graphs are Eulerian [16]. In another direction, research focuses on full self-twuals for regular maps and when they may be self-dual, self-petrial, or self-Wilsonial, [5,12,21].…”

“…[1, 5, −6, 2, 3, 7, −6, −4, 3, 5, −4, 1, 2, 7] = (δτ, δ, τ δτ, τ δ, τ δτ, 1, τ δ) H 7 (1, 6, 3) 7 [−1, −2, 7][−1, 4, 5, 3, 7, −6, 4, 3, −2, −6, 5] = (τ, δ, 1, τ δ, τ δτ, τ, τ δ) H 7 (1, 6, 3) 7 [−1, −5, 6, 4, −5, 3, −2, 6, −7, 3, 4, −1, −2, −7] = (τ δτ, δ, τ δ, δτ, τ δτ, τ δτ, τ δ) H 7 (1, 6, 3) 7 [1, 5, −6, 2, 3, 7, −6, 4, 1, 2, 7][3, 5, 4] = (δτ, δ, τ δτ, 1, τ δτ, 1, τ δ) H 7 (1, 6, 3) 7 [−1, 5, 4, −1, −2, −7][−2, −6, −7, 3, 4, −6, 5, 3]= (τ δτ, δ, τ δ, τ δ, δ, τ δτ, τ δ) H 7(1,6,3) …”

Partial Duality of Hypermaps

“…Since then partial duality was thoughtfully studied and described in many papers [6,13,16,17,18,20,23,24,24,25,26,27,28]. For an excellent exposition see [14].…”

“…Proposition 3.2 Partial duality relative to an edge for gems [7,13]. Let e be an edge of a ribbon graph G and C be the corresponding 02-cycle of its gem.…”